Tubed incompressible surfaces in knot and link complements
The main result of the paper shows that if a non-split link in \(S^3\) has a plat projection satisfying certain conditions, then its complement contains closed incompressible non-boundary-parallel surfaces. Moreover, when the link is a knot these surfaces remain incompressible in all manifolds obtained by nontrivial surgery on the knot. More precisely, one considers a \(2n\)-plat projection of a non-split link, that is, a \(2n\)-braid with the \((2j-1)\)st and \((2j)\)th strands connected by arcs at the top and bottom, for \(1\leq j\leq n\), and with the braiding crossings collected into \(m\) rows where in the \(i\)th row there are \(a_{ij}\) crossings between the \((2j)\)th and \((2j+1)\)st strands for \(1\leq j\leq n-1\) if \(i\) is odd, and between the \((2j-1)\)st and \((2j)\)th strands for \(1\leq j\leq n\) if \(i\) is even. Here, \(a_{ij}\) is positive or negative according as the crossings are positive or negative. The result is that if \(n\geq 3\), \(m\geq 5\), and \(|a_{ij}|>2\) for all \(i\) and \(j\), then the complement of the link contains closed incompressible surfaces of genus \((m+1)/2\). Moreover, if the link is a knot, then all manifolds obtained by nontrivial Dehn surgery are irreducible and contain incompressible surfaces of genus \((m+1)/2\). The idea of the proof is to start with spheres that follow near the \((2j-1)\)st and \((2j)\)th strands, cutting across the link in \(m+1\) points. By cutting out small discs near the intersection points and adding tubes that follow along the link, a closed surface of genus \((m+1)/2\) is obtained. To show that it is incompressible, the link is decomposed into tangles in a precise way, and a hypothetical compressing disc for the surface is carefully isotoped out of each of the tangles until it can be seen to be parallel into the surface. The argument is complicated; most of the work occurs in the case of \(n=3\), which furnishes the starting point for induction on \(n\). The additional conclusions for the case of a knot are obtained using results of W. Menasco and M. Scharlemann.
scientific article; zbMATH DE number 1340402